The non-global part \(S_\text{non-global}^{\mathcal {B}_\delta }(t)\)  is
computed numerically, in the large-\(N_c\)  limit, following the algorithm
highlighted in [1]}, as also done
in [2]}.Note that our definition
of the scaled variable \(t(L)\)  differs from the one
in [1]} by a factor 2 and from the one in
e.g. [4]}, [5]}, [6]}, [7]}
by factor of \(1/2\) .
This can straightforwardly be applied to ungroomed angularities.
For SoftDrop groomed distributions, the non-global factor
remains the same for \(v\ge z_\text{cut}\)  and saturates at that value,
i.e. \(S_\text{non-global}^\text{(groomed)}(v) =S_\text{non-global}^\text{(ungroomed)}(\text{max}(v,z_\text{cut}))\) .
We also note that the non-global contribution only depends on \(v\)  (and
the jet transverse momentum through \(\alpha _s(\mu _R)\) ) but not on the angularity
parameter \(\alpha \) .
Similarly, for the groomed case, the contribution from non-global logarithms only depends on \(z_\text{cut}\)  and
not on \(\beta \) .
The main reason behind this is that, at single-logarithmic accuracy, non-global logarithms originate
from configurations that are strongly ordered in energy with angles
commensurate with the jet radius \(R_0\) .
